This implementation of an Inductive (recursive) data type involves the categorical co-product construction. See: LimitsAndColimits

(1) -> )library colimits

Reading /var/aw/var/LatexWiki/colimits.asy
   Sum2 is now explicitly exposed in frame initial 
   Sum2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/colimits
   Sum2Functions is now explicitly exposed in frame initial 
   Sum2Functions will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/colimits
   Sum3 is now explicitly exposed in frame initial 
   Sum3 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/colimits
   Sum3Functions is now explicitly exposed in frame initial 
   Sum3Functions will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/colimits

An example of Arithmetic Expressions as an Inductive Type

#pile
#include "axiom"
#library COLIMIT "colimits.ao"
import from COLIMIT
+++ Category of Integer-valued Arthemetic Expresions

ExprCat:Category == with
    eval: % -> Integer
    -- evaluates the expression as an Integer
    coerce: % -> OutputForm
    -- displays the expression

+++ Constructor for atomic Integers
MkInt(Z:IntegerNumberSystem): ExprCat
  == add
    Rep == Integer
    -- Export
    eval(x:%):Integer == rep(x)
    coerce(x:%):OutputForm == outputForm rep x

+++ Constructor for additions
MkAdd(X:ExprCat,Y:ExprCat): ExprCat
  == add
    Rep==Record(left:X,right:Y)
    import from Rep, OutputForm
    -- Local:
    plus(x:X,y:Y):OutputForm == paren(x::OutputForm + y::OutputForm);
    sum(x:X,y:Y):Integer     == eval(x) + eval(y);
    -- Export:
    eval(x:%):Integer == sum explode rep x
    coerce(x:%):OutputForm == plus explode rep x

+++ Constructor for multiplications
MkMul(X:ExprCat,Y:ExprCat): ExprCat
  == add
    Rep==Record(left:X,right:Y)
    import from Rep, OutputForm
    -- Local:
    times(x:X,y:Y):OutputForm== paren(x::OutputForm * y::OutputForm);
    product(x:X,y:Y):Integer == eval(x) * eval(y);
    -- Export:
    eval(x:%):Integer == product explode rep x
    coerce(x:%):OutputForm == times explode rep x

MI==>MkInt(Integer)
MA==>MkAdd(%,%)
MM==>MkMul(%,%)

+++ Constructor of Arithmetic Expressions over the Integers
Expr: ExprCat
  == add
    Rep == Sum(MI,MA,MM)
    import from Rep, MI, MA, MM
    -- Export:
    eval(x:%):Integer == sum(Integer,eval,eval,eval)(rep x)
    coerce(x:%):OutputForm == sum(OutputForm,coerce,coerce,coerce)(rep x)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/expr.as using
      Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/expr.as", line 2: 
#include "axiom"
^
[L2 C1] #1 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

Now extend it to provide the Axiom interpreter interface.

#pile
#include "axiom"
#library EXPR "expr.ao"
import from EXPR
#library COLIMIT "colimits.ao"
import from COLIMIT

extend MkInt(Z:IntegerNumberSystem): with
    coerce: Z -> %
    -- converts a Z-valued object to an atomic Integer
  == add
    Rep == Integer
    -- Export
    coerce(i:Z):% == per(convert(i))

extend MkAdd(X:ExprCat,Y:ExprCat): with
    +: (X,Y) -> %
    -- returns an expression representing the sum
  == add
    Rep==Record(left:X,right:Y)
    import from Rep
    -- Export
    ((x:X) + (y:Y)):% == per [x,y]

extend MkMul(X:ExprCat,Y:ExprCat): with
    *: (X,Y) -> %
    -- returns an expression representing the product
  == add
    Rep==Record(left:X,right:Y)
    import from Rep
    -- Export
    ((x:X) * (y:Y)):% == per [x,y]

MI==>MkInt(Integer)
MA==>MkAdd(%,%)
MM==>MkMul(%,%)

+++ extends domain Expr as above
extend Expr:  with
    coerce: Integer -> %
    +:(%,%) -> %
    *:(%,%) -> %
  == add
    Rep == Sum(MI,MA,MM)
    import from Rep, MI, MA, MM
    -- Export:
    coerce(n: Integer):% == per(inject( n::MI ))
    ((x:%) + (y:%)):% ==    per(inject( (x + y)$MA ))
    ((x:%) * (y:%)):% ==    per(inject( (x * y)$MM ))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1062037838131250571-25px003.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1062037838131250571-25px003.as", line 2: 
#include "axiom"
^
[L2 C1] #1 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

n:Expr:=3

   Expr is not a valid type.

Ralf Hemmecke and Bill Page wrote:

Recursive Types

#include "axiom"
RecExpr:with {
    coerce: Integer -> %;
    +: (%, %) -> %;
    *: (%, %) -> %;
    coerce: % -> OutputForm;
    eval: % -> Integer;
}  == add {
    Rep == Union(MakeInt:Integer,
                 MakeAdd:Record(l:%, r:%),
                 MakeMul:Record(l:%, r:%));
    import from Rep;
    import from 'MakeInt', 'MakeAdd', 'MakeMul';
    import from OutputForm;
    --
    -- Local
    plus(x:%,y:%):OutputForm == paren(x::OutputForm + y::OutputForm);
    times(x:%,y:%):OutputForm== paren(x::OutputForm * y::OutputForm);
    sum(x:%,y:%):Integer     == eval(x) + eval(y);
    product(x:%,y:%):Integer == eval(x) * eval(y);
    --
    -- Input
    coerce(n: Integer):% == per [MakeInt==n];
    ((x:%) + (y:%)): %   == per [MakeAdd==[x,y]];
    ((x:%) * (y:%)): %   == per [MakeMul==[x,y]];
    --
    -- Output
    coerce(i:%):OutputForm == { n:=rep(i);
      n case MakeInt => outputForm    n.MakeInt;
      n case MakeAdd => plus  explode n.MakeAdd;
      n case MakeMul => times explode n.MakeMul;
      never;
    };
    --
    -- Evaluation
    eval(i:%):Integer == { n:=rep(i);
      n case MakeInt =>                 n.MakeInt;
      n case MakeAdd => sum     explode n.MakeAdd;
      n case MakeMul => product explode n.MakeMul;
      never;
    }
}

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4075514935174965511-25px005.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4075514935174965511-25px005.as", line 1: 
#include "axiom"
^
[L1 C1] #1 (Error) Could not open file `axiom'.

   The )library system command was not called after compilation.

Here is an example expression

10::RecExpr

   RecExpr is not a valid type.